# Analysis of random non-autonomous logistic-type differential equations via the Karhunen-Lo\`eve expansion and the Random Variable Transformation technique

## TL;DR

This paper develops a probabilistic approach to analyze logistic differential equations with uncertain initial conditions and stochastic diffusion coefficients, using Karhunen-Loève expansion and variable transformation to approximate the solution's probability density.

## Contribution

It introduces a novel method combining Karhunen-Loève expansion and variable transformation to approximate the probability density function of solutions to stochastic logistic equations.

## Key findings

- Proves convergence of the approximation as the expansion order increases
- Provides explicit formulas for the approximate density functions
- Demonstrates the method with three illustrative examples

## Abstract

This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, $f_1(p,t)$, of the solution stochastic process, $P(t,\omega)$. To achieve this goal, first the diffusion coefficient is represented via a truncation of order $N$ of the Karhunen-Lo\`{e}ve expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximations, say $f_1^N(p,t)$, of $f_1(p,t)$ are constructed. Afterwards, we rigorously prove that $f_1^N(p,t) \longrightarrow f_1(p,t)$ as $N\to \infty$ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.